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Elliptic problems with superlinear convection terms

L. Boccardo, S. Buccheri, G. R. Cirmi

Abstract

In this manuscript we deal with elliptic equations with superlinear first order terms in divergence form of the following type \[ -\mbox{div}(M(x)\nabla u)= -\mbox{div}(h(u)E(x))+f(x), \] where $M$ is a bounded elliptic matrix, the vector field $E$ and the function $f$ belong to suitable Lebesgue spaces, and the function $s\to h(s)$ features a superlinear growth at infinity. We provide some existence and non existence results for solutions to the associated Dirichlet problem and a comparison principle.

Elliptic problems with superlinear convection terms

Abstract

In this manuscript we deal with elliptic equations with superlinear first order terms in divergence form of the following type where is a bounded elliptic matrix, the vector field and the function belong to suitable Lebesgue spaces, and the function features a superlinear growth at infinity. We provide some existence and non existence results for solutions to the associated Dirichlet problem and a comparison principle.
Paper Structure (6 sections, 11 theorems, 130 equations)

This paper contains 6 sections, 11 theorems, 130 equations.

Table of Contents

  1. Introduction and main results
  2. Bibliographic background
  3. Proof of the results
  4. Slightly superlinear case
  5. Strongly superlinear case
  6. The impact of the zero order term

Key Result

Theorem 1.1

Assume alfa, ipoh, growthg and take $E\in [L^{r}(\Omega)]^N$, with $r>N$, and $f\in L^{m}(\Omega)$, with $m>\frac{N}{2}$. Then, there exists a unique weak solution $u\in W^{1,2}_0(\Omega)\cap L^{\infty}(\Omega)$ to problem problem.

Theorems & Definitions (26)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • ...and 16 more